Electromagnetically induced transparency of ultralong-range Rydberg molecules
We study the impact of Rydberg molecule formation on the storage and retrieval of Rydberg polaritons in an ultracold atomic medium. We observe coherent revivals appearing in the retrieval efficiency of stored photons that originate from simultaneous excitation of Rydberg atoms and Rydberg molecules in the system with subsequent interference between the possible storage paths. We show that over a large range of principal quantum numbers the observed results can be described by a two-state model including only the atomic Rydberg state and the Rydberg dimer molecule state. At higher principal quantum numbers the influence of polyatomic molecules becomes relevant and the dynamics of the system undergoes a transition from coherent evolution of a few-state system to an effective dephasing into a continuum of molecular states.
Mapping the long-range interaction between Rydberg atoms Saffman et al. (2010) onto slowly traveling polaritons via electromagnetically induced transparency (EIT) Fleischhauer et al. (2005) has emerged as a promising approach to realize effective photon-photon interaction in an optical medium Friedler et al. (2005); Shahmoon et al. (2011); Pritchard et al. (2010); Gorshkov et al. (2011); Peyronel et al. (2012). Fast development over the last years in this novel field of Rydberg quantum optics Firstenberg et al. (2016) has enabled new tools for quantum information as well as for exploring dynamics of correlated quantum many-body systems Chang et al. (2014), including efficient single-photon generation Dudin and Kuzmich (2012); Maxwell et al. (2013), creation of entanglement between light and atomic excitations Li et al. (2013), realization of attractive forces between single photons Firstenberg et al. (2013), demonstration of single-photon all-optical switches Baur et al. (2014) and transistors Gorniaczyk et al. (2014); Tiarks et al. (2014); Gorniaczyk et al. (2016), single-photon absorbers Tresp et al. (2016) and interaction induced photon phase shifts Parigi et al. (2012); Tiarks et al. (2016); Thompson et al. (2017). Future prospects include the crystallization of photons Gorshkov et al. (2013); Otterbach et al. (2013) and the observation of three-body interaction between photons Jachymski et al. (2016); Gullans et al. (2016).
The critical figure of merit for most of the above work is the optical depth (OD) per blockade volume Gorshkov et al. (2011). Improving this quantity requires increasing the atomic density of the medium, but this inevitably brings the system into the regime where formation of ultralong-range Rydberg molecules Greene et al. (2000a); Bendkowsky et al. (2009) has to be taken into account to describe the Rydberg-polariton dynamics. The experimental study of these exotic molecules has evolved into a highly active field in itself, with a variety of exciting phenomena having been explored so far, such as states bound by quantum reflection Bendkowsky et al. (2010), coherent creation and breaking of the molecular bond Butscher et al. (2011), polyatomic Rydberg molecules Gaj et al. (2014), trilobite Li et al. (2011); Booth et al. (2015) and butterfly Niederprüm et al. (2016a) states, and controlled hybridization of the molecular bond Gaj et al. (2015). Diatomic Rydberg molecules have been realized for -states in Rb Bendkowsky et al. (2009), Cs Tallant et al. (2012) and Sr DeSalvo et al. (2015), for -states in Rb Anderson et al. (2014a); Krupp et al. (2014) and for -states in Rb Bellos et al. (2013) and Cs Saßmannshausen et al. (2015). Furthermore, Rb Rydberg molecules have been used as a probe of the quantum phase transition from the superfluid to the Mott-insulator phase Manthey et al. (2015) and the existence of Rydberg molecules bound by mixed singlet-triplet scattering Anderson et al. (2014b) has been proven experimentally for Cs Saßmannshausen et al. (2015) and Rb Böttcher et al. (2016); Niederprüm et al. (2016b).
In the context of Rydberg quantum optics, formation of Rydberg molecules has been suggested as a limiting factor on the coherence of slow and stored Rydberg polaritons Gaj et al. (2014). Experimentally, Dürr et al. have observed a dephasing rate of single photons stored in a Rydberg state with large principal quantum number () increasing linearly with atomic density, restricting the overall performance of an all-optical switch Baur et al. (2014).
In the present work, we study the storage of Rydberg polaritons systematically over a large range of principal quantum numbers and atomic densities to verify the connection between polariton dephasing and molecule formation. Specifically, we are able to match the binding energy of Rydberg dimers to the period in an observed revival of the retrieval efficiency in an intermediate range of principal quantum numbers . Our results are well-described by a simple two-species model, including Rydberg atoms and Rydberg dimers, from which we extract the coherence time of photons stored in Rydberg dimers. At higher principal quantum numbers, we observe the transition from coherent dynamics to an effective dephasing when the number of involved molecular states grows Gaj et al. (2014); Baur et al. (2014). Experiments and analysis similar to ours have recently been performed by Baur et al. Baur (2015).
A schematic of our experiment is shown in Fig. 1. We prepare Rb atoms, pumped into the ground state , trapped in a crossed optical dipole trap. The radial and axial radii of the cigar shaped cloud at the temperature of are and . A weak probe field, which couples the ground state to the intermediate state , is focused onto the center of the cloud (). We measure on the transition over the full cloud length. For coupling the probe photons to a Rydberg state we add a strong counterpropagating control field (). To measure Rydberg excitation spectra we detune both probe and control fields by from the intermediate state and then scan the two-photon detuning over the to two-photon transition. After the excitation we field-ionize Rydberg atoms and collect ions on a microchannel plate detector (MCP). To perform storage and retrieval experiments, we employ the sequence shown in Fig. 1(c). First, we send a Gaussian probe pulse containing on average photons into the cloud under resonant conditions () and ramp the intensity of the control field down to zero on a timescale of while the probe pulse propagates through the cloud. As a result, a part of the probe pulse is stored as a stopped Rydberg polariton. To read out the Rydberg polariton from the cloud, the control field is turned back on after variable storage time and the output optical pulse is collected on a single-photon counting module (SPCM).
We start our studies by investigating photon storage and Rydberg molecule formation over a range of principal quantum numbers between . Fig. 2 shows the observed retrieval efficiency plotted versus storage time for four different Rydberg states. We observe oscillations in the retrieval efficiency for , and with frequencies of , and . To relate these oscillations to Rydberg dimers, we show the excitation spectra in the corresponding insets in Fig. 2. Each spectrum consists of an atomic peak corresponding to excitation of Rydberg atoms, situated at the origin, and a clearly distinguishable dimer line red-detuned from the atomic line. The underlying mechanism for Rydberg molecule formation is the scattering of the slow Rydberg electron from a ground state atom Greene et al. (2000b). In our case, the observed Rydberg dimers are well-described by pure s-wave scattering Bendkowsky et al. (2009), resulting in a molecular binding potential
where is the local electron density and is the s-wave scattering length for the electron - Rb atom collision Gaj et al. (2014). The vibrational ground state wavefunction of the dimer is mainly localized in the outermost lobe of the Rydberg electron wavefunction, resulting in a characteristic scaling of the dimer binding energy with principal quantum number Bendkowsky et al. (2009). The Rydberg electron can also capture multiple ground state atoms, forming polyatomic molecules with binding energy of a molecule with ground state atoms times larger than the binding energy of the dimer Gaj et al. (2014). In our experiment the mean atomic density over the full cloud is , which is sufficient to form a significant fraction of Rydberg dimers, but only a negligible fraction of polyatomic Rydberg molecules for . We extract the Rydberg dimer binding energy (in the following we will refer to as a binding energy in frequency units) by fitting a sum of two Gaussian lineshapes to the spectra, finding , and for , and correspondingly, which are in good agreement with the measured retrieval oscillation frequencies. For , the dimer fraction is very small while the dimer binding energy is larger than the Rydberg EIT linewidth, which we set to FWHM for all datasets. As expected, in this case no oscillations in retrieval efficiency are observed (Fig. 2(a)).
To quantitatively reproduce the measured retrieval curves we employ a simple two-species model for the stored Rydberg polaritons. We consider our atomic cloud to consist of Rydberg atoms, positioned such that they cannot form a Rydberg molecule, and pairs of atoms, positioned such that they can form a Rydberg dimer. For Rydberg states where the dimer binding energy is smaller than the EIT linewidth , each photon is stored in a superposition of collective Rydberg atom and dimer excitations. The wave function of the stored polariton then evolves in time as
Here, , is a collective state with atom j in Rydberg state and all others in ground state , is the symmetrized dimer state with atom pair j forming a Rydberg molecule. After a storage time the retrieval efficiency is then proportional to the overlap of this wave function with the original one at
In addition to the coherent evolution, we take into account several dephasing mechanisms. First, the thermal atomic motion introduces an overall Gaussian decay of the retrieval efficiency with lifetime , where is the sum of the wavevectors of the EIT light fields and is the thermal speed of atoms. Additionally, we take into account spontaneous decay of both the Rydberg atoms and Rydberg dimers with lifetimes and respectively. Including these effects in eq. 3, we write the time dependence of the retrieval efficiency of a stored polariton as
where is the initial retrieval efficiency which is determined experimentally; and are the fractions of Rydberg atoms and Rydberg dimers, normalized such that . The oscillations in the retrieval efficiency emerge from eq. 4 because of the interference between the polariton component stored in the atomic fraction with the one stored in the dimer fraction. We note that because we send on average photons per experimental realization, we rarely store more than one photon and thus can neglect any interaction between stored polaritons Bariani et al. (2012); Dudin and Kuzmich (2012); Busche et al. (2017).
We use eq. 4 to fit the experimental data in Fig. 2. The Rydberg atom lifetime can be calculated and therefore we fix it during the fitting to (), (), () and (). First, for the state (Fig. 2(a)), we set , which leaves only the thermal dephasing as a free parameter for which we obtain . This value is consistent with the dephasing time inferred from the cloud temperature measured by absorption imaging, which suggests of . Thus the time dependence of the retrieval efficiency, when we do not couple to molecular states, is well described by decay due to random thermal motion of atoms and Rydberg population decay.
Next, for the data shown in Fig. 2(b)-(d), we keep , , and as free parameters. The resulting values for are in very good agreement with the values we get from the excitation spectra (see above). Lifetimes of thermal atomic motion extracted from the fits (), () and (), all of which are again in very good agreement with the measured cloud temperature. The parameters defining the contrast of the observed oscillations, are the fraction of Rydberg atoms and the fraction of Rydberg dimers . For our set of data, the fit yields a Rydberg fraction of 0.89(1) (), 0.82(1) () and 0.75(1) (). When we compare these values to the ones obtained from the amplitudes of the Gaussian fits to atomic and molecular lines in the excitation spectra, 0.80(1) (), 0.73(1) (), 0.75(2) (), we observe discrepancies for the and states, exceeding one standard deviation. It is important to note that the result of the fit is very sensitive to the value of and it is not possible to compensate an offset from the optimum value of by adjusting other fit parameters. Instead, the observed disagreement in the parameters is explained by systematic drifts of the total atom number, which over the time required for the two measurements is on the order of 10% in our setup. For the shown datasets we can confirm these drifts by evaluating control measurements of optical depth taken throughout the full measurement sequence, which confirm the change in atom number between the spectra and retrieval data sets. Finally, we extract the Rydberg dimer coherence times (), () and (). These values are comparable to the Rydberg dimer lifetimes measured by other methods in previous experiments Butscher et al. (2010, 2011); Weber et al. (2015). While our method does not differentiate between decay time and dephasing time Butscher et al. (2010), our results suggest that coherent photon storage in the Rydberg dimer state is possible over time scales required for conditional photon-photon interaction Baur et al. (2014); Gorniaczyk et al. (2016); Tiarks et al. (2016).
We next turn to investigating the time dependence of the retrieval efficiency for different atomic densities, which allows to alter the Rydberg dimer fraction . Fig. 3 shows the measured retrieval efficiency curves at four atomic densities for the state as well as the corresponding Rydberg excitation spectra. We extract the mean atomic density for each dataset from the position of the center of gravity of the excitation spectrum Gaj et al. (2014), obtaining , , and (Fig. 3(e)-(h)). Correspondingly, the Rydberg dimer fraction shrinks with the density down to for the lowest density. Therefore, as expected from eq. 4, we observe a decreasing contrast in the retrieval efficiency oscillations when lowering the density (Fig. 3(a)-(d)). More quantitatively, by fitting the data with the model, we extract the Rydberg atom fractions for the four datasets, which are again in agreement with the ratios extracted from the excitation spectra within our experiment stability. In Fig. 3(i) we show the Rydberg dimer coherence times extracted from the fits for the different atomic densities. Omitting the result for the lowest density, where the dimer fraction is very small and the oscillations are barely visible, the remaining data points suggest a linear decrease of the coherence time as was found previously by Baur et al. Baur et al. (2014).
Next, we investigate photon storage in a Rydberg state with larger principal quantum number, specifically , to highlight the transition from coherent dynamics to effective dephasing observed previously at Baur et al. (2014). For the state the dimer lineshift is already smaller than the dimer linewidth, resulting in an asymmetric Rydberg line in the excitation spectrum with a non-resolvable shoulder on the red-detuned side (Fig. 4(b)). From the spectrum, we can thus no longer directly determine the dimer binding energy. This can only be done indirectly by determining the scaling of the binding energy with effective quantum number from the data at lower (Fig. 4(c)). Fitting a power law to the binding energies extracted for , , and , we obtain , which is in very good agreement with the value found by Gaj et al. Gaj et al. (2014). Using the binding energy predicted from this scaling, we mark the line centers of the atomic, dimer, trimer and tetramer contributions in the spectrum shown in Fig. 4(c). From these line positions it becomes clear that for and our atomic density , there should already be a sizable contribution from Rydberg trimers and even tetramers in the photon storage process. Consequently, our two-species model fails to adequately describe the observed revival in the retrieval efficiency in this case (dashed violet line in Fig. 4(a)). We can significantly improve our model by adding terms corresponding to Rydberg trimers (dashed green line) and tetramers (red line) with binding energies multiples of the dimer binding energy to eq. 4. The fit with a total of four species (atoms, dimers, trimers, tetramers) reproduces the single revival feature very well, adding further polymer states does not improve the fit. Importantly, from this fit we obtain the dimer binding energy as (grey triangle in Fig. 4(c)), in very good agreement with the value predicted by the scaling law. This shows that the photon storage and retrieval is a useful method to measure Rydberg molecule binding energies even if the molecular lines are not resolvable in the excitation spectrum.
In conclusion, we have investigated the coherent oscillations appearing in the retrieval efficiency of stored photons in experiments on storage and retrieval of Rydberg polaritons. We explain these oscillations by simultaneous excitation of Rydberg atoms and Rydberg dimers in the system with subsequent interference between the possible storage paths. Our observations are well reproduced by a simple model including only the Rydberg dimer state over a range of principal quantum numbers . For higher principal quantum numbers, more molecular states become relevant. We show that our model still works well for if the trimer and tetramer vibrational ground states are included. At even higher principal quantum number, the density of molecular states increases so much, that the retrieval evolution is well described by a single decay term Baur et al. (2014). A more sophisticated model would require including larger polyatomic molecules as well as vibrational excited states of the Rydberg molecules Schmidt et al. (2016). From the perspective of Rydberg quantum optics, our observation of coherent light storage in Rb Rydberg dimers with principal quantum numbers close to the two-state Förster resonance exploited for efficient all-optical switch and transistor operation Tiarks et al. (2014); Gorniaczyk et al. (2016); Paris-Mandoki et al. (2016) should prove highly relevant for the further improvement of these devices and for the realization of coherent photonic gates based on Rydberg-mediated photon-photon interaction.
Acknowledgements.We acknowledge funding by the German Research Foundation (Emmy-Noether-grant HO 4787/1-1, GiRyd project HO 4787/1-3, SFB/TRR21 project C12) and the Ministry of Science, Research and the Arts of Baden-Württemberg (RiSC grant 33-7533.-30-10/37/1)
- Saffman et al. (2010) M. Saffman, T. G. Walker, and K. Mølmer, Rev. Mod. Phys. 82, 2313 (2010).
- Fleischhauer et al. (2005) M. Fleischhauer, A. Imamoğlu, and J. Marangos, Rev. Mod. Phys. 77, 633 (2005).
- Friedler et al. (2005) I. Friedler, D. Petrosyan, M. Fleischhauer, and G. Kurizki, Phys. Rev. A 72, 043803 (2005).
- Shahmoon et al. (2011) E. Shahmoon, G. Kurizki, M. Fleischhauer, and D. Petrosyan, Phys. Rev. A 83, 033806 (2011).
- Pritchard et al. (2010) J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, Phys. Rev. Lett. 105, 193603 (2010).
- Gorshkov et al. (2011) A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, Phys. Rev. Lett. 107, 133602 (2011).
- Peyronel et al. (2012) T. Peyronel, O. Firstenberg, Q. Liang, S. Hofferberth, A. Gorshkov, T. Pohl, M. Lukin, and V. Vuletić, Nature 488, 57 (2012).
- Firstenberg et al. (2016) O. Firstenberg, C. S. Adams, and S. Hofferberth, Journal of Physics B: Atomic, Molecular and Optical Physics 49, 152003 (2016).
- Chang et al. (2014) D. E. Chang, V. Vuletić, and M. D. Lukin, Nature Photonics 8, 685 (2014).
- Dudin and Kuzmich (2012) Y. O. Dudin and A. Kuzmich, Science 336, 887 (2012).
- Maxwell et al. (2013) D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, Phys. Rev. Lett. 110, 103001 (2013).
- Li et al. (2013) L. Li, Y. O. Dudin, and A. Kuzmich, Nature 498, 466 (2013).
- Firstenberg et al. (2013) O. Firstenberg, T. Peyronel, Q. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletić, Nature 502, 71 (2013).
- Baur et al. (2014) S. Baur, D. Tiarks, G. Rempe, and S. Dürr, Phys. Rev. Lett. 112, 073901 (2014).
- Gorniaczyk et al. (2014) H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, Phys. Rev. Lett. 113, 053601 (2014).
- Tiarks et al. (2014) D. Tiarks, S. Baur, K. Schneider, S. Dürr, and G. Rempe, Phys. Rev. Lett. 113, 053602 (2014).
- Gorniaczyk et al. (2016) H. Gorniaczyk, C. Tresp, P. Bienias, A. Paris-Mandoki, W. Li, I. Mirgorodskiy, H. P. Büchler, I. Lesanovsky, and S. Hofferberth, Nature Communications 7, 12480 (2016).
- Tresp et al. (2016) C. Tresp, C. Zimmer, I. Mirgorodskiy, H. Gorniaczyk, A. Paris-Mandoki, and S. Hofferberth, Phys. Rev. Lett. 117, 223001 (2016).
- Parigi et al. (2012) V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, Phys. Rev. Lett. 109, 233602 (2012).
- Tiarks et al. (2016) D. Tiarks, S. Schmidt, G. Rempe, and S. Dürr, Science Advances 2, e1600036 (2016).
- Thompson et al. (2017) J. D. Thompson, T. L. Nicholson, Q.-Y. Liang, S. H. Cantu, A. V. Venkatramani, S. Choi, I. A. Fedorov, D. Viscor, T. Pohl, M. D. Lukin, and V. Vuletić, Nature 542, 206 (2017).
- Gorshkov et al. (2013) A. V. Gorshkov, R. Nath, and T. Pohl, Phys. Rev. Lett. 110, 153601 (2013).
- Otterbach et al. (2013) J. Otterbach, M. Moos, D. Muth, and M. Fleischhauer, Phys. Rev. Lett. 111, 113001 (2013).
- Jachymski et al. (2016) K. Jachymski, P. Bienias, and H. P. Büchler, Phys. Rev. Lett. 117, 053601 (2016).
- Gullans et al. (2016) M. J. Gullans, J. D. Thompson, Y. Wang, Q.-Y. Liang, V. Vuletić, M. D. Lukin, and A. V. Gorshkov, Phys. Rev. Lett. 117, 113601 (2016).
- Greene et al. (2000a) C. H. Greene, A. S. Dickinson, and H. R. Sadeghpour, Phys. Rev. Lett. 85, 2458 (2000a).
- Bendkowsky et al. (2009) V. Bendkowsky, B. Butscher, J. Nipper, J. P. Shaffer, R. Löw, and T. Pfau, Nature 458, 1005 (2009).
- Bendkowsky et al. (2010) V. Bendkowsky, B. Butscher, J. Nipper, J. B. Balewski, J. P. Shaffer, R. Löw, T. Pfau, W. Li, J. Stanojevic, T. Pohl, and J. M. Rost, Phys. Rev. Lett. 105, 163201 (2010).
- Butscher et al. (2011) B. Butscher, V. Bendkowsky, J. Nipper, J. B. Balewski, L. Kukota, R. Löw, T. Pfau, W. Li, T. Pohl, and J. M. Rost, Journal of Physics B: Atomic, Molecular and Optical Physics 44, 184004 (2011).
- Gaj et al. (2014) A. Gaj, A. T. Krupp, J. B. Balewski, R. Löw, S. Hofferberth, and T. Pfau, Nature communications 5, 4546 (2014).
- Li et al. (2011) W. Li, T. Pohl, J. M. Rost, S. T. Rittenhouse, H. R. Sadeghpour, J. Nipper, B. Butscher, J. B. Balewski, V. Bendkowsky, R. Löw, and T. Pfau, Science 334, 1110 (2011).
- Booth et al. (2015) D. Booth, S. T. Rittenhouse, J. Yang, H. R. Sadeghpour, and J. P. Shaffer, Science 348, 99 (2015).
- Niederprüm et al. (2016a) T. Niederprüm, O. Thomas, T. Eichert, C. Lippe, J. PÃ©rez-RÃos, C. H. Greene, and H. Ott, Nature Communications 7, 12820 (2016a).
- Gaj et al. (2015) A. Gaj, A. T. Krupp, P. Ilzhöfer, R. Löw, S. Hofferberth, and T. Pfau, Phys. Rev. Lett. 115, 023001 (2015).
- Tallant et al. (2012) J. Tallant, S. T. Rittenhouse, D. Booth, H. R. Sadeghpour, and J. P. Shaffer, Phys. Rev. Lett. 109, 173202 (2012).
- DeSalvo et al. (2015) B. J. DeSalvo, J. A. Aman, F. B. Dunning, T. C. Killian, H. R. Sadeghpour, S. Yoshida, and J. Burgdörfer, Phys. Rev. A 92, 031403 (2015).
- Anderson et al. (2014a) D. A. Anderson, S. A. Miller, and G. Raithel, Phys. Rev. A 90, 062518 (2014a).
- Krupp et al. (2014) A. T. Krupp, A. Gaj, J. B. Balewski, P. Ilzhöfer, S. Hofferberth, R. Löw, T. Pfau, M. Kurz, and P. Schmelcher, Phys. Rev. Lett. 112, 143008 (2014).
- Bellos et al. (2013) M. A. Bellos, R. Carollo, J. Banerjee, E. E. Eyler, P. L. Gould, and W. C. Stwalley, Phys. Rev. Lett. 111, 053001 (2013).
- Saßmannshausen et al. (2015) H. Saßmannshausen, F. Merkt, and J. Deiglmayr, Phys. Rev. A 92, 032505 (2015).
- Manthey et al. (2015) T. Manthey, T. Niederprüm, O. Thomas, and H. Ott, New Journal of Physics 17, 103024 (2015).
- Anderson et al. (2014b) D. A. Anderson, S. A. Miller, and G. Raithel, Phys. Rev. A 90, 062518 (2014b).
- Böttcher et al. (2016) F. Böttcher, A. Gaj, K. M. Westphal, M. Schlagmüller, K. S. Kleinbach, R. Löw, T. C. Liebisch, T. Pfau, and S. Hofferberth, Phys. Rev. A 93, 032512 (2016).
- Niederprüm et al. (2016b) T. Niederprüm, O. Thomas, T. Eichert, and H. Ott, Phys. Rev. Lett. 117, 123002 (2016b).
- Baur (2015) S. E. Baur, A Single-Photon Switch and Transistor based on Rydberg Blockade, Dissertation, Technische Universität München, München (2015).
- Greene et al. (2000b) C. H. Greene, A. S. Dickinson, and H. R. Sadeghpour, Phys. Rev. Lett. 85, 2458 (2000b).
- Bariani et al. (2012) F. Bariani, Y. O. Dudin, T. A. B. Kennedy, and A. Kuzmich, Phys. Rev. Lett. 108, 030501 (2012).
- Busche et al. (2017) H. Busche, P. Huillery, S. W. Ball, T. Ilieva, M. P. A. Jones, and C. S. Adams, Nature Physics (2017), doi:10.1038/nphys4058.
- Butscher et al. (2010) B. Butscher, J. Nipper, J. B. Balewski, L. Kukota, V. Bendkowsky, R. Löw, and T. Pfau, Nature Physics 6, 970 (2010).
- Weber et al. (2015) T. M. Weber, M. Höning, T. Niederprüm, T. Manthey, O. Thomas, V. Guarrera, M. Fleischhauer, G. Barontini, and H. Ott, Nature Physics 11, 157 (2015).
- Schmidt et al. (2016) R. Schmidt, H. R. Sadeghpour, and E. Demler, Phys. Rev. Lett. 116, 105302 (2016).
- Paris-Mandoki et al. (2016) A. Paris-Mandoki, H. Gorniaczyk, C. Tresp, I. Mirgorodskiy, and S. Hofferberth, Journal of Physics B: Atomic, Molecular and Optical Physics 49, 164001 (2016).